How should "planet" be defined?
The current official
IAU definition of planet doesn't really make sense for multiple reasons.
0. Objects can be "exoplanet" or dwarf planet" without actually being a planet, that simply doesn't make any grammatical sense.
1. Whether or not an object is a planet depends on it's orbit/position.
The least problematic criteria is with regard to hydrostatic equilibrium but even then it would be better to go on mass.
The biggest problem with going only on mass is that it's often hard to measure that. With Sedna for example we can only guess the mass based on the density of pluto and it's size but we currently do not have anything close to an actual measurements of it's mass.
With exoplanets however we sometimes get a more accurate figure for its mass since we discover it by looking at how it gravitationally affects its nearest star.
One way to differentiate moons + planet systems from planet + planet systems is to use the following
it's a binary system if the barycenter of the objects' orbits is not within one of the objects. That's the line between binary system and planetary system. In that case, neither object is a moon and they co-dominate their orbit, so they'd both be a part of a binary planet system.
That is currently not followed strictly since Charon is often described as a moon of pluto even though the barycenter is outside Pluto. We need to follow that more strictly.
My suggestion is to use the following to determine what counts as major moons/planets and what counts as minor moons/planets.
Major moon/planet: Any object with radius of at least 240 Km in all directions from it's centre of mass, mass more than 2*10^22 and mass smaller than 2.4 * 10^28 Kg
Minor moon/planet: Any object with radius of at least 240 Km and mass less than 2 * 10^22 Kg
Satelite Asteroid: sufficiently large natural sattelite that
That would result in the following 8 major planets
Terrestrial planets
☿ Mercury
♀ Venus
🜨 Earth
♂ Mars
Gas giants
♃ Jupiter
♄ Saturn
⛢ Uranus
♆ Neptune
You would also get 7 major moons
The
Moon orbiting Earth.
Io,
Europa,
Ganymede, and
Callisto orbiting Jupiter.
Titan orbiting saturn.
Triton orbiting Neptune.
With the following moons being in the minor category regardless
Mimas,
Enceladus,
Tethys,
Dione,
Rhea, and
Iapetus orbiting Saturn.
Miranda,
Ariel,
Umbriel,
Titania, and
Oberon orbiting Uranus.
https://en.wikipedia.org/wiki/List_of_Solar_System_objects_by_size
Minor planets aka dwarf planets
Ceres
Orcus
Pluto
Charon technically in a binary system with Pluto.
Haumea
Quaoar
Makemake
Gonggong
Eris
Sedna
Salacia
Note that you would basically get the same outcome by having a minimum diameter of 1200Km for something to be considered a "major moon/planet" since none of the moons/planets smaller diameter than that has mass above 20*10^21 Kg.